Geometric Mean

Insert two geometric means between $9$ and $243$.

Insert four geometric means between $6$ and $192$.

Insert three geometric means between $\frac{1}{3}$ and $432$.

Find two positive numbers $a$ and $b$, whose

$AM=10$ and $GM=8$

Find two positive numbers $a$ and $b$, whose

$AM=25$ and $GM=20$

For what values of $x$ are the numbers $\frac{-2}{7},x,\frac{-7}{2}$ in GP?

If $AM$ and $GM$ of the roots of a quadratic equation are $10$ and $8$ respectively then obtain the quadratic equation.

Show that the product of $n$ geometric means between $a$ and $b$ is equal to the $n^{th}$ power of the single $GM$ between $a$ and $b$.

If $a, b, c$ are in $\mathrm{AP}, x$ is the $\mathrm{GM}$ between $a$ and $b; y$ is the $\mathrm{GM}$ between $b$ and $c;$ then show that $b^2$ is the $\mathrm{AM}$ between $x^2$ and $y^2$.

The $\mathrm{AM}$ between two positive numbers $a$ and $b(a>b)$ is twice their $\mathrm{GM}$. Prove that $a:b=(2+\sqrt{3}):(2-\sqrt{3})$.

The $GM$ between the numbers$a^{3}b \mathrm{and} a{b}^3$ is $a^k{b}^k$. Find the value of $k$.

Find the $GM$ between the numbers

$-6.3$ and $-2.8$

Find the $GM$ between the numbers

$-8$ and $-2$

Find the $GM$ between the numbers

$0.15$ and $0.0015$

The geometric mean of $1$ and $\frac{9}{16}$ is $\frac{k}{4}$.  Find the values of $k$.

Find the $GM$ between the numbers

$5$ and $125$

If $a, b, c$ are in $GP$ and $a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{2}}$ then prove that $x, y, z$ are in $AP.$

Write the quadratic equation, the arithmetic and geometric means of whose roots are $A$ and $G$ respectively.

If $(a-b), (b-c), (c-a)$ are in $\mathrm{GP}$ then prove that $(a+b+c{)}^2=3(ab+bc+ca)$.

If $a, b, c$ are in $\mathrm{GP},$ prove that $\frac{a^2+ab+b^2}{ab+bc+ca}=\frac{b+a}{c+b}$